Randomized mesh design

ABSTRACT

In one embodiment, an apparatus may include a touch sensor that includes a mesh of conductive material. The mesh includes a number of mesh cells that each have a number of vertices. Each of the vertices has a substantially randomized location within an annulus centered at a seed location of the vertex. The apparatus may also include one or more computer-readable non-transitory storage media coupled to the touch sensor and embodying logic that is configured when executed to control the touch sensor.

TECHNICAL FIELD

This disclosure generally relates to touch sensors.

BACKGROUND

An array of conductive drive and sense electrodes may form amutual-capacitance touch sensor having one or more capacitive nodes. Themutual-capacitance touch sensor may have either a two-layerconfiguration or single-layer configuration. In a single-layerconfiguration, drive and sense electrodes may be disposed in a patternon one side of a substrate. In such a configuration, a pair of drive andsense electrodes capacitively coupled to each other across a space ordielectric between electrodes may form a capacitive node.

In a single-layer configuration for a self-capacitance implementation,an array of vertical and horizontal conductive electrodes may bedisposed in a pattern on one side of the substrate. Each of theconductive electrodes in the array may form a capacitive node, and, whenan object touches or comes within proximity of the electrode, a changein self-capacitance may occur at that capacitive node and a controllermay measure the change in capacitance as a change in voltage or a changein the amount of charge needed to raise the voltage to somepre-determined amount.

In a touch-sensitive display application, a touch screen may enable auser to interact directly with what is displayed on a display underneaththe touch screen, rather than indirectly with a mouse or touchpad. Atouch screen may be attached to or provided as part of, for example, adesktop computer, laptop computer, tablet computer, personal digitalassistant (PDA), smartphone, satellite navigation device, portable mediaplayer, portable game console, kiosk computer, point-of-sale device, orother suitable device. A control panel on a household or other appliancemay include a touch screen.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an example touch sensor with an example controller.

FIG. 2 illustrates example mesh cells with example vertices havingsubstantially randomized locations.

FIG. 3 illustrates example dual-layer mesh cells with example verticeshaving substantially randomized locations.

FIG. 4 illustrates an example placement of example seed locationsrelative to an example display.

FIG. 5 illustrates an example method for designing a conductive meshwith randomized vertices.

FIG. 6 illustrates an example computer system.

DESCRIPTION OF EXAMPLE EMBODIMENTS

FIG. 1 illustrates an example touch sensor 10 with an exampletouch-sensor controller 12. Touch sensor 10 and touch-sensor controller12 may detect the presence and location of a touch or the proximity ofan object within a touch-sensitive area of touch sensor 10. Herein,reference to a touch sensor may encompass both the touch sensor and itstouch-sensor controller, where appropriate. Similarly, reference to atouch-sensor controller may encompass both the touch-sensor controllerand its touch sensor, where appropriate. Touch sensor 10 may include oneor more touch-sensitive areas, where appropriate. Touch sensor 10 mayinclude an array of drive and sense electrodes (or an array ofelectrodes of a single type) disposed on one or more substrates, whichmay be made of a dielectric material. Herein, reference to a touchsensor may encompass both the electrodes of the touch sensor and thesubstrate(s) that they are disposed on, where appropriate.Alternatively, where appropriate, reference to a touch sensor mayencompass the electrodes of the touch sensor, but not the substrate(s)that they are disposed on.

An electrode (whether a ground electrode, a guard electrode, a driveelectrode, or a sense electrode) may be an area of conductive materialforming a shape, such as for example a disc, square, rectangle, thinline, other suitable shape, or suitable combination of these. One ormore cuts in one or more layers of conductive material may (at least inpart) create the shape of an electrode, and the area of the shape may(at least in part) be bounded by those cuts. In particular embodiments,the conductive material of an electrode may occupy approximately 100% ofthe area of its shape. As an example and not by way of limitation, anelectrode may be made of an optically clear conductive material, such asfor example indium tin oxide (ITO) and the ITO of the electrode mayoccupy approximately 100% of the area of its shape (sometimes referredto as 100% fill), where appropriate. In particular embodiments, theconductive material of an electrode may occupy substantially less than100% of the area of its shape. As an example and not by way oflimitation, an electrode may be made of fine lines of metal or otherconductive material (FLM), such as for example copper, silver, or acopper- or silver-based material, and the fine lines of conductivematerial may occupy approximately 5% of the area of its shape in ahatched, mesh, or other suitable pattern. Herein, reference to FLMencompasses such material, where appropriate. Although this disclosuredescribes or illustrates particular electrodes made of particularconductive material forming particular shapes with particular fillpercentages having particular patterns, this disclosure contemplates anysuitable electrodes made of any suitable conductive material forming anysuitable shapes with any suitable fill percentages having any suitablepatterns.

Where appropriate, the shapes of the electrodes (or other elements) of atouch sensor may constitute in whole or in part one or moremacro-features of the touch sensor. One or more characteristics of theimplementation of those shapes (such as, for example, the conductivematerials, fills, or patterns within the shapes) may constitute in wholeor in part one or more micro-features of the touch sensor. One or moremacro-features of a touch sensor may determine one or morecharacteristics of its functionality, and one or more micro-features ofthe touch sensor may determine one or more optical features of the touchsensor, such as transmittance, refraction, or reflection.

A mechanical stack may contain the substrate (or multiple substrates)and the conductive material forming the drive or sense electrodes oftouch sensor 10. As an example and not by way of limitation, themechanical stack may include a first layer of optically clear adhesive(OCA) beneath a cover panel. The cover panel may be clear and made of aresilient material suitable for repeated touching, such as for exampleglass, polycarbonate, or poly(methyl methacrylate) (PMMA). Thisdisclosure contemplates any suitable cover panel made of any suitablematerial. The first layer of OCA may be disposed between the cover paneland the substrate with the conductive material forming the drive orsense electrodes. The mechanical stack may also include a second layerof OCA and a dielectric layer (which may be made of PET or anothersuitable material, similar to the substrate with the conductive materialforming the drive or sense electrodes). As an alternative, whereappropriate, a thin coating of a dielectric material may be appliedinstead of the second layer of OCA and the dielectric layer. The secondlayer of OCA may be disposed between the substrate with the conductivematerial making up the drive or sense electrodes and the dielectriclayer, and the dielectric layer may be disposed between the second layerof OCA and an air gap to a display of a device including touch sensor 10and touch-sensor controller 12. As an example only and not by way oflimitation, the cover panel may have a thickness of approximately 1 mm;the first layer of OCA may have a thickness of approximately 0.05 mm;the substrate with the conductive material forming the drive or senseelectrodes may have a thickness of approximately 0.05 mm; the secondlayer of OCA may have a thickness of approximately 0.05 mm; and thedielectric layer may have a thickness of approximately 0.05 mm. Althoughthis disclosure describes a particular mechanical stack with aparticular number of particular layers made of particular materials andhaving particular thicknesses, this disclosure contemplates any suitablemechanical stack with any suitable number of any suitable layers made ofany suitable materials and having any suitable thicknesses. As anexample and not by way of limitation, in particular embodiments, a layerof adhesive or dielectric may replace the dielectric layer, second layerof OCA, and air gap described above, with there being no air gap to thedisplay.

One or more portions of the substrate of touch sensor 10 may be made ofpolyethylene terephthalate (PET) or another suitable material. Thisdisclosure contemplates any suitable substrate with any suitableportions made of any suitable material. In particular embodiments, thedrive or sense electrodes in touch sensor 10 may be made of ITO in wholeor in part. In particular embodiments, the drive or sense electrodes intouch sensor 10 may be made of fine lines of metal or other conductivematerial. As an example and not by way of limitation, one or moreportions of the conductive material may be copper or copper-based andhave a thickness of approximately 5 μm or less and a width ofapproximately 10 μm or less. As another example, one or more portions ofthe conductive material may be silver or silver-based and similarly havea thickness of approximately 5 μm or less and a width of approximately10 μm or less. This disclosure contemplates any suitable electrodes madeof any suitable material.

Touch sensor 10 may implement a capacitive form of touch sensing. In amutual-capacitance implementation, touch sensor 10 may include an arrayof drive and sense electrodes forming an array of capacitive nodes. Adrive electrode and a sense electrode may form a capacitive node. Thedrive and sense electrodes forming the capacitive node may come neareach other, but not make electrical contact with each other. Instead,the drive and sense electrodes may be capacitively coupled to each otheracross a space between them. A pulsed or alternating voltage applied tothe drive electrode (by touch-sensor controller 12) may induce a chargeon the sense electrode, and the amount of charge induced may besusceptible to external influence (such as a touch or the proximity ofan object). When an object touches or comes within proximity of thecapacitive node, a change in capacitance may occur at the capacitivenode and touch-sensor controller 12 may measure the change incapacitance. By measuring changes in capacitance throughout the array,touch-sensor controller 12 may determine the position of the touch orproximity within the touch-sensitive area(s) of touch sensor 10.

In a self-capacitance implementation, touch sensor 10 may include anarray of electrodes of a single type that may each form a capacitivenode. When an object touches or comes within proximity of the capacitivenode, a change in self-capacitance may occur at the capacitive node andtouch-sensor controller 12 may measure the change in capacitance, forexample, as a change in the amount of charge needed to raise the voltageat the capacitive node by a pre-determined amount. As with amutual-capacitance implementation, by measuring changes in capacitancethroughout the array, touch-sensor controller 12 may determine theposition of the touch or proximity within the touch-sensitive area(s) oftouch sensor 10. This disclosure contemplates any suitable form ofcapacitive touch sensing, where appropriate.

In particular embodiments, one or more drive electrodes may togetherform a drive line running horizontally or vertically or in any suitableorientation. Similarly, one or more sense electrodes may together form asense line running horizontally or vertically or in any suitableorientation. In particular embodiments, drive lines may runsubstantially perpendicular to sense lines. Herein, reference to a driveline may encompass one or more drive electrodes making up the driveline, and vice versa, where appropriate. Similarly, reference to a senseline may encompass one or more sense electrodes making up the senseline, and vice versa, where appropriate.

Touch sensor 10 may have drive and sense electrodes disposed in apattern on one side of a single substrate. In such a configuration, apair of drive and sense electrodes capacitively coupled to each otheracross a space between them may form a capacitive node. For aself-capacitance implementation, electrodes of only a single type may bedisposed in a pattern on a single substrate. In addition or as analternative to having drive and sense electrodes disposed in a patternon one side of a single substrate, touch sensor 10 may have driveelectrodes disposed in a pattern on one side of a substrate and senseelectrodes disposed in a pattern on another side of the substrate.Moreover, touch sensor 10 may have drive electrodes disposed in apattern on one side of one substrate and sense electrodes disposed in apattern on one side of another substrate. In such configurations, anintersection of a drive electrode and a sense electrode may form acapacitive node. Such an intersection may be a location where the driveelectrode and the sense electrode “cross” or come nearest each other intheir respective planes. The drive and sense electrodes do not makeelectrical contact with each other—instead they are capacitively coupledto each other across a dielectric at the intersection. Although thisdisclosure describes particular configurations of particular electrodesforming particular nodes, this disclosure contemplates any suitableconfiguration of any suitable electrodes forming any suitable nodes.Moreover, this disclosure contemplates any suitable electrodes disposedon any suitable number of any suitable substrates in any suitablepatterns.

As described above, a change in capacitance at a capacitive node oftouch sensor 10 may indicate a touch or proximity input at the positionof the capacitive node. Touch-sensor controller 12 may detect andprocess the change in capacitance to determine the presence and locationof the touch or proximity input. Touch-sensor controller 12 may thencommunicate information about the touch or proximity input to one ormore other components (such one or more central processing units (CPUs))of a device that includes touch sensor 10 and touch-sensor controller12, which may respond to the touch or proximity input by initiating afunction of the device (or an application running on the device).Although this disclosure describes a particular touch-sensor controllerhaving particular functionality with respect to a particular device anda particular touch sensor, this disclosure contemplates any suitabletouch-sensor controller having any suitable functionality with respectto any suitable device and any suitable touch sensor.

Touch-sensor controller 12 may be one or more integrated circuits (ICs),such as for example general-purpose microprocessors, microcontrollers,programmable logic devices or arrays, application-specific ICs (ASICs).In particular embodiments, touch-sensor controller 12 comprises analogcircuitry, digital logic, and digital non-volatile memory. In particularembodiments, touch-sensor controller 12 is disposed on a flexibleprinted circuit (FPC) bonded to the substrate of touch sensor 10, asdescribed below. The FPC may be active or passive, where appropriate. Inparticular embodiments, multiple touch-sensor controllers 12 aredisposed on the FPC. Touch-sensor controller 12 may include a processorunit, a drive unit, a sense unit, and a storage unit. The drive unit maysupply drive signals to the drive electrodes of touch sensor 10. Thesense unit may sense charge at the capacitive nodes of touch sensor 10and provide measurement signals to the processor unit representingcapacitances at the capacitive nodes. The processor unit may control thesupply of drive signals to the drive electrodes by the drive unit andprocess measurement signals from the sense unit to detect and processthe presence and location of a touch or proximity input within thetouch-sensitive area(s) of touch sensor 10. The processor unit may alsotrack changes in the position of a touch or proximity input within thetouch-sensitive area(s) of touch sensor 10. The storage unit may storeprogramming for execution by the processor unit, including programmingfor controlling the drive unit to supply drive signals to the driveelectrodes, programming for processing measurement signals from thesense unit, and other suitable programming, where appropriate. Althoughthis disclosure describes a particular touch-sensor controller having aparticular implementation with particular components, this disclosurecontemplates any suitable touch-sensor controller having any suitableimplementation with any suitable components.

Tracks 14 of conductive material disposed on the substrate of touchsensor 10 may couple the drive or sense electrodes of touch sensor 10 toconnection pads 16, also disposed on the substrate of touch sensor 10.As described below, connection pads 16 facilitate coupling of tracks 14to touch-sensor controller 12. Tracks 14 may extend into or around (e.g.at the edges of) the touch-sensitive area(s) of touch sensor 10.Particular tracks 14 may provide drive connections for couplingtouch-sensor controller 12 to drive electrodes of touch sensor 10,through which the drive unit of touch-sensor controller 12 may supplydrive signals to the drive electrodes. Other tracks 14 may provide senseconnections for coupling touch-sensor controller 12 to sense electrodesof touch sensor 10, through which the sense unit of touch-sensorcontroller 12 may sense charge at the capacitive nodes of touch sensor10. Tracks 14 may be made of fine lines of metal or other conductivematerial. As an example and not by way of limitation, the conductivematerial of tracks 14 may be copper or copper-based and have a width ofapproximately 100 μm or less. As another example, the conductivematerial of tracks 14 may be silver or silver-based and have a width ofapproximately 100 μm or less. In particular embodiments, tracks 14 maybe made of ITO in whole or in part in addition or as an alternative tofine lines of metal or other conductive material. Although thisdisclosure describes particular tracks made of particular materials withparticular widths, this disclosure contemplates any suitable tracks madeof any suitable materials with any suitable widths. In addition totracks 14, touch sensor 10 may include one or more ground linesterminating at a ground connector (which may be a connection pad 16) atan edge of the substrate of touch sensor 10 (similar to tracks 14).

Connection pads 16 may be located along one or more edges of thesubstrate, outside the touch-sensitive area(s) of touch sensor 10. Asdescribed above, touch-sensor controller 12 may be on an FPC. Connectionpads 16 may be made of the same material as tracks 14 and may be bondedto the FPC using an anisotropic conductive film (ACF). Connection 18 mayinclude conductive lines on the FPC coupling touch-sensor controller 12to connection pads 16, in turn coupling touch-sensor controller 12 totracks 14 and to the drive or sense electrodes of touch sensor 10. Inanother embodiment, connection pads 16 may be connected to anelectro-mechanical connector (such as a zero insertion forcewire-to-board connector); in this embodiment, connection 18 may not needto include an FPC. This disclosure contemplates any suitable connection18 between touch-sensor controller 12 and touch sensor 10.

FIG. 2 illustrates example mesh cells with example vertices havingsubstantially randomized locations. Although this disclosure describesand illustrates a particular distribution of seed locations, thisdisclosure contemplates any suitable distribution of seed locations.Moreover, although this disclosure describes and illustrates particularvertices defining particular mesh cells or microfeatures in particularconfigurations, this disclosure contemplates any suitable verticesdefining any suitable mesh cells or microfeatures in any suitableconfiguration. Area 20 may correspond to a portion of a drive or senseelectrode (or other element) of a touch sensor. In a touch sensor, meshsegments 70 connecting pairs of adjacent vertices 74 may correspond tofine lines of metal (such as for example copper, silver, or a copper- orsilver-based material) or other conductive material with a thickness ofapproximately 1 μm or less and a width of approximately 5 μm or less.Seed locations 72, on the other hand, do not correspond to anyconductive or other material in the touch sensor. Instead, they mayserve as a basis to determine at least in part the arrangement ofvertices 74, as described below. In particular embodiments, mesh cells76 may be defined at least in part by two pairs of opposing vertices 74and associated mesh segments 70. Although this disclosure describes andillustrates particular mesh cells with a particular number andconfiguration of vertices and mesh segments, this disclosurecontemplates any suitable mesh cell with any suitable number of verticesand mesh segments.

In particular embodiments, seed locations 72 may be distributedthroughout area 20 in a two-dimensional (2D) substantially regularlyspaced pattern. In particular embodiments, seed locations 72 may bedistributed based at least in part on vertices 74 of an initial meshcell (e.g. 76A). As an example and not by way of limitation, seedlocations 72 of the initial mesh cell (e.g. 76A) may have an initialdistribution. As described below, vertices 74 of the initial mesh cell(e.g. 76A) may be determined through an annulus 78 of each vertex 74 ofthe initial mesh cell. Furthermore, seed locations 72 of subsequent meshcells (e.g. 76B) may be determined based at least in part on thevertices 74 of the initial mesh cell (e.g. 76A). Although thisdisclosure describes and illustrates particular distribution of seedlocations, this disclosure contemplates any suitable distribution ofseed locations, such as for example a substantially random distribution.

Vertices 74 of mesh cells 76A-B may be arranged in a substantiallyrandomized pattern that may reduce the occurrence of repeating patternsor frequencies among mesh segments 70, which may in turn reduce theoccurrence of moiré patterns with respect to a display visible througharea 20. In particular embodiments, each seed location 72 may have anassociated annulus 78 substantially centered about each seed location 72and annulus 78 may be defined by an associated a minimum 80 and maximum82 pre-determined radii. As an example and not by way of limitation, adimension of minimum 80 and maximum 82 pre-determined radii may bedetermined based at least in part of one or more dimensions of andisplay underneath area 20. In particular embodiments, a location ofeach vertex 74 may be substantially randomly distributed within theannulus 78 associated with each seed location 72. Furthermore, meshsegments 70 of conductive material may couple adjacent pairs of vertices74 as described above.

FIG. 3 illustrates an example dual-layer mesh pattern with substantiallyrandomized vertices. The example of FIG. 2 illustrates a single-sidedimplementation, but this disclosure contemplates any suitable n-sidedimplementation and is not limited to a singled-sided implementation. Asdescribed above, area 20 may correspond to a portion of a drive or senseelectrode (or other element) of a touch sensor. In particularembodiments, a dual-layer mesh pattern over area 20 may include a secondmesh of conductive material separated from a first mesh of conductivematerial at least by a thickness of a dielectric layer. As an exampleand not by way of limitation, a first conductive mesh may be formed on afirst substrate and a second conductive mesh may be formed on a secondsubstrate. As another example, the first and second conductive meshesmay be formed on a surface of a substrate with a layer of dielectricmaterial at locations where one or more mesh segments of the secondconductive mesh overlap a mesh segment of the first conductive mesh.Furthermore, the first conductive mesh may correspond to at least aportion of a drive electrode and the second conductive mesh maycorrespond to at least a portion of a sense electrode of a touch sensoror vice versa.

In particular embodiments, seed locations 88, and therefore one or morevertices 74B, of the second conductive mesh may be distributed based atleast in part on the location of mesh cells of the first conductivemesh. As an example and not by way of limitation, the distribution ofseed locations 88 of the second conductive mesh may be based at least inpart on a centroid of mesh cells of the first conductive mesh defined byvertices 74 as illustrated in the example of FIG. 3. As described above,each seed location 88 of the second conductive mesh may have anassociated annulus 78. As an example and not by way of limitation, theradii of annuli 78 associated with seed locations 88 may besubstantially equal to the pre-determined radii of the annuli,illustrated in the example of FIG. 2, of the first conductive mesh. Inparticular embodiments, a location of each vertex 74B of the secondconductive mesh may be substantially randomly distributed within theannulus 78 associated with each seed location 88.

As described in regard to the example of FIG. 2, mesh segments 70B ofconductive material may couple adjacent pairs of vertices 74B of thesecond conductive mesh. In particular embodiments, one or more meshcells 86A of the second conductive mesh may be formed by couplingadjacent pairs of vertices 74B with a minimum length mesh segment 70B.In particular embodiments, one or more mesh cells 86B of the secondconductive mesh may be formed by coupling adjacent pairs of vertices 74Bwith one or more mesh segments 70B that overlap a mid-point location 84of a mesh segment of the first conductive mesh, thereby forming amulti-segmented coupling between adjacent pairs of vertices 74B.

FIG. 4 illustrates an example placement of example seed locationsrelative to an example display. A touch sensor may be overlaid on adisplay to implement a touch-sensitive display device, as describedbelow. As an example and not by way of limitation, the displayunderneath the touch sensor may be a liquid crystal display (LCD), alight-emitting diode (LED) display, an LED backlight LCD, anelectrophoretic display, a plasma display, or other suitable display.Although this disclosure describes and illustrates a particular displayconfiguration and display type, this disclosure contemplates anysuitable device display configuration and display type. An exampleportion of a display may include an array of pixels 22. In the exampleof FIG. 4, each pixel 22 may include three sub-pixels 24. In particularembodiments, each sub-pixel 24 may correspond to a particular color,such as for example red, green, or blue. Furthermore, the combinedoutput of sub-pixels 24 may determine the color and intensity of eachpixel 22. Although this disclosure describes and illustrates pixels witha particular number of sub-pixels having particular colors, thisdisclosure contemplates any suitable pixels with any suitable number ofsub-pixels having any suitable colors.

In particular embodiments, pixels 22 or sub-pixels 24 may be arranged ina repeating pattern along a horizontal axis 28 and a vertical axis 32that are perpendicular to each other. Although this disclosure describesand illustrates horizontal and vertical axes, this disclosurecontemplates any suitable axes having any suitable orientation. Eachpixel 22 has a horizontal-pixel pitch (HPP) 26, which in particularembodiments may be defined as the distance between correspondingfeatures of two adjacent pixels 22 along horizontal axis 28 (such as thehorizontal distance from the left edge of two horizontally adjacentsub-pixels 24). Each pixel 22 has a vertical-pixel pitch (VPP) 30, whichin particular embodiments may be defined as the distance betweencorresponding features of two adjacent pixels 22 along vertical axis 32(such as the vertical distance from the lower edge of two verticallyadjacent sub-pixels 24). This disclosure contemplates any suitablepixels with any suitable HPPs and VPPs having any suitable values.

Each pixel 22 may also include a dead space, which corresponds toregions of pixel 22 not occupied by a sub-pixel 24. In particularembodiments, the dead space has a height 34 that may be defined as thedistance between adjacent sub-pixels 24 along vertical axis 32 (such asthe distance between two vertically adjacent sub-pixels 24). Inparticular embodiments, the dead space has a width 36 that may bedefined as the distance between adjacent sub-pixels 24 along horizontalaxis 28 (such as the distance between two horizontally adjacentsub-pixels 24). This disclosure contemplates any suitable pixels withany suitable dead space having any suitable dimensions. Each sub-pixel24 has a horizontal sub-pixel pitch (HSPP) 38, which may be defined inparticular embodiments as the distance between corresponding features oftwo adjacent sub-pixels along horizontal axis 28, including width 36 ofthe dead space (such as the distance between the left edges of twohorizontally adjacent sub-pixels 24). Each sub-pixel 24 also has avertical sub-pixel pitch (VSPP) 40, which may be defined in particularembodiments as the distance between corresponding features of twoadjacent sub-pixels along vertical axis 32, including height 34 of thedead space (such as the distance between the lower edges of twovertically adjacent sub-pixels 24).

Each sub-pixel 24 has a sub-pixel width (SPW) 42, which may be definedin particular embodiments as the dimension of a sub-pixel alonghorizontal axis 28 (such as the distance between the left and rightedges of sub-pixel 24). Each sub-pixel 24 also has a sub-pixel height(SPH) 44, which is defined in particular embodiments as the dimension ofa sub-pixel along vertical axis 32 (such as the distance between thelower and upper edges of sub-pixel 24). This disclosure contemplates anysuitable sub-pixels with any suitable HSPPs, VSPPs, SPWs, and SPHshaving any suitable values. In particular embodiments, pixel 22 andsub-pixel 24 may have a substantially rectangular shape, as illustratedin the example of FIG. 4. Pixel 22 and sub-pixel 24 may have othersuitable shape, such as for example square, round, oval, orchevron-shaped. In the example of FIG. 4, vertical-sub-pixel pitch 40 isequal to VPP 30, and VPP 30 is equal to the sum of SPH 44 and dead-spaceheight 34. Further, HPP 26 is equal to three times HSPP 38, and HSPP 38is equal to the sum of sub-pixel width 42 and dead-space width 36.Although this disclosure describes and illustrates example pixels 22 andexample sub-pixels 24 having particular shapes, arrangements, anddimensions, this disclosure contemplates any suitable arrangement of anysuitable pixels and sub-pixels having any suitable shapes anddimensions.

In particular embodiments, seed locations (e.g. 72A-C) may be determinedbased at least in part on one or more dimensions of a display underneaththe touch sensor. In the example of FIG. 4, a portion of seed locations(e.g. 72A-B) may be linearly disposed along a terminal side of an angle54 relative to horizontal axis 28, and another portion of seed locations(e.g. 72B-C) may be linearly disposed along a terminal side 52 of anangle 56 relative to horizontal axis 28. In particular embodiments, theterminal side 50 of angle 54 may pass through points 58 and 60. Theslope of terminal side 50 of angle 54 may be defined as the verticalrise of terminal side 50 divided by the horizontal run of terminal side50, and angle 54 can be found from the arctangent of the slope. In theexample of FIG. 4, the vertical rise of terminal side 50 is SPH 44, andthe horizontal run of terminal side 50 is HPP 26. Thus, the slope ofterminal side 50 equals SPH/HPP, and angle 54 (Θ₁) can be found from theexpression Θ₁=arctan(SPH/HPP).

In the example of FIG. 4, terminal side 52 of angle 56 associated with aportion of seed locations (e.g. 72C-B) may pass through points 62 and64. The slope of terminal side 52 of angle 56 may be defined as thevertical rise of terminal side 52 divided by the horizontal run ofterminal side 52, and angle 56 can be found from the arctangent of theslope. In the example of FIG. 4, the vertical rise of terminal side 52is VPP 30, and the horizontal run of terminal side 52 is two times HSPP38. Thus, the slope of terminal side 52 equals VPP/2·HSPP, and angle 56(Θ₂) can be found from the expression Θ₂=arctan(VPP/2·HSPP). Inparticular embodiments, angles Θ₁ and Θ₂ may vary by up to approximately1° from the values calculated in the expressions above withoutsubstantially degrading the optical properties associated with the meshpattern. Terminal sides 50 and 52, and points 58, 60, 62, and 64 do notcorrespond to any conductive or other material in the touch sensor andare provided for didactic purposes. Furthermore, terminal side 50 ofangle 54 need not be constrained to pass through particular points 58and 60 or seed locations 72A-B illustrated in the example of FIG. 4, butmay be displaced along horizontal axis 28 and vertical axis 32 by anysuitable amount. Similarly, in particular embodiments, terminal side 52of angle 56 need not be constrained to pass through points 62 and 64 orseed locations 72B-C as illustrated in the example of FIG. 4, but may bedisplaced along horizontal axis 28 and vertical axis 32 by any suitableamount.

As an example and not by way of limitation, a horizontal distance 90separating terminal side 50 of angle 54 passing through seed locations72A and 72B, from terminal side 50 of angle 54 passing through seedlocations 72C and 72D may be substantially equal to three times HPP 26(or nine times HSPP 38). A horizontal distance 92 separating terminalside 52 of angle 56 passing through seed locations 72B and 72C, fromterminal side 52 of angle 56 passing through seed locations 72A and 72Emay be substantially equal to 13/6 times HPP 26 (or 6.5 times HSPP 38).Furthermore, horizontal distances 90 and 92 described above may becompatible with a display with an HPP of approximately 150 μm. Asanother example, horizontal distance 90 separating terminal side 50 ofangle 54 passing through seed locations 72A and 72B, from terminal side50 of angle 54 passing through seed locations 72C and 72D may besubstantially equal to six times HPP 26 (or 18 times HSPP 38).Horizontal distance 92 separating terminal side 52 of angle 56 passingthrough seed locations 72B and 72C, from terminal side 52 of angle 56passing through seed locations 72A and 72E may be substantially equal to13/3 times HPP 26 (or 13 times HSPP 38). Furthermore, horizontaldistances 90 and 92 described above may be compatible for a display withan HPP that is substantially less than 150 μm. As another example,horizontal distance 90 separating terminal side 50 of angle 54 passingthrough seed locations 72A and 72B, from terminal side 50 of angle 54passing through seed locations 72C and 72D may be substantially equalsubstantially equal to two times HPP 26 (or six times HSPP 38).Horizontal distance 92 separating terminal side 52 of angle 56 passingthrough seed locations 72B and 72C, from terminal side 52 of angle 56passing through seed locations 72A and 72E may be substantially equal tothe sum of HPP 26, HSPP 38, dead space width 36, and ½ of sub-pixelwidth 42. Horizontal distances 90 and 92 described above may becompatible with a display with an HPP 26 of approximately 250 μm.Although this disclosure describes and illustrates particular horizontaldistances between particular seed locations, this disclosurecontemplates any suitable separation distances between any suitable seedlocations.

FIG. 5 illustrates an example method for designing a conductive meshwith randomized vertices. The method may start at step 100, where acomputing device may determine a number of seed locations. In particularembodiments, the seed locations may be a regularly spaced 2D patternthat may be determined at least in part on one or more dimensions of adisplay. At step 102, the computing device may generate a pattern for amesh of conductive material of a touch sensor at least in part bydetermining a number of vertices of a number of mesh cells of the meshof conductive material, at which point the method may end. In particularembodiments, each of the vertices may have a substantially randomizedlocation within an annulus centered at one of the seed locations.Although this disclosure describes and illustrates particular steps ofthe method of FIG. 5 as occurring in a particular order, this disclosurecontemplates any suitable steps of the method of FIG. 5 occurring in anysuitable order. Particular embodiments may repeat one or more steps ofthe method of FIG. 4, where appropriate. Moreover, although thisdisclosure describes and illustrates an example method for designing aconductive mesh with randomized vertices including the particular stepsof the method of FIG. 5, this disclosure contemplates any suitablemethod for designing a conductive mesh with randomized verticesincluding any suitable steps, which may include all, some, or none ofthe steps of the method of FIG. 5, where appropriate. Moreover, althoughthis disclosure describes and illustrates particular components carryingout particular steps of the method of FIG. 5, this disclosurecontemplates any suitable combination of any suitable componentscarrying out any suitable steps of the method of FIG. 5.

FIG. 6 illustrates an example computer system 200. In particularembodiments, one or more computer systems 200 perform one or more stepsof one or more methods described or illustrated herein. In particularembodiments, one or more computer systems 200 provide functionalitydescribed or illustrated herein. In particular embodiments, softwarerunning on one or more computer systems 200 performs one or more stepsof one or more methods described or illustrated herein or providesfunctionality described or illustrated herein. Particular embodimentsinclude one or more portions of one or more computer systems 200.Herein, reference to a computer system may encompass a computing device,and vice versa, where appropriate. Moreover, reference to a computersystem may encompass one or more computer systems, where appropriate.

This disclosure contemplates any suitable number of computer systems200. This disclosure contemplates computer system 200 taking anysuitable physical form. As example and not by way of limitation,computer system 200 may be an embedded computer system, a system-on-chip(SOC), a single-board computer system (SBC) (such as, for example, acomputer-on-module (COM) or system-on-module (SOM)), a desktop computersystem, a laptop or notebook computer system, an interactive kiosk, amainframe, a mesh of computer systems, a mobile telephone, a personaldigital assistant (PDA), a server, a tablet computer system, or acombination of two or more of these. Where appropriate, computer system200 may include one or more computer systems 200; be unitary ordistributed; span multiple locations; span multiple machines; spanmultiple data centers; or reside in a cloud, which may include one ormore cloud components in one or more networks. Where appropriate, one ormore computer systems 200 may perform without substantial spatial ortemporal limitation one or more steps of one or more methods describedor illustrated herein. As an example and not by way of limitation, oneor more computer systems 200 may perform in real time or in batch modeone or more steps of one or more methods described or illustratedherein. One or more computer systems 200 may perform at different timesor at different locations one or more steps of one or more methodsdescribed or illustrated herein, where appropriate.

In particular embodiments, computer system 200 includes a processor 202,memory 204, storage 206, an input/output (I/O) interface 208, acommunication interface 210, and a bus 212. Although this disclosuredescribes and illustrates a particular computer system having aparticular number of particular components in a particular arrangement,this disclosure contemplates any suitable computer system having anysuitable number of any suitable components in any suitable arrangement.

In particular embodiments, processor 202 includes hardware for executinginstructions, such as those making up a computer program. As an exampleand not by way of limitation, to execute instructions, processor 202 mayretrieve (or fetch) the instructions from an internal register, aninternal cache, memory 204, or storage 206; decode and execute them; andthen write one or more results to an internal register, an internalcache, memory 204, or storage 206. In particular embodiments, processor202 may include one or more internal caches for data, instructions, oraddresses. This disclosure contemplates processor 202 including anysuitable number of any suitable internal caches, where appropriate. Asan example and not by way of limitation, processor 202 may include oneor more instruction caches, one or more data caches, and one or moretranslation lookaside buffers (TLBs). Instructions in the instructioncaches may be copies of instructions in memory 204 or storage 206, andthe instruction caches may speed up retrieval of those instructions byprocessor 202. Data in the data caches may be copies of data in memory204 or storage 206 for instructions executing at processor 202 tooperate on; the results of previous instructions executed at processor202 for access by subsequent instructions executing at processor 202 orfor writing to memory 204 or storage 206; or other suitable data. Thedata caches may speed up read or write operations by processor 202. TheTLBs may speed up virtual-address translation for processor 202. Inparticular embodiments, processor 202 may include one or more internalregisters for data, instructions, or addresses. This disclosurecontemplates processor 202 including any suitable number of any suitableinternal registers, where appropriate. Where appropriate, processor 202may include one or more arithmetic logic units (ALUs); be a multi-coreprocessor; or include one or more processors 202. Although thisdisclosure describes and illustrates a particular processor, thisdisclosure contemplates any suitable processor.

In particular embodiments, memory 204 includes main memory for storinginstructions for processor 202 to execute or data for processor 202 tooperate on. As an example and not by way of limitation, computer system200 may load instructions from storage 206 or another source (such as,for example, another computer system 200) to memory 204. Processor 202may then load the instructions from memory 204 to an internal registeror internal cache. To execute the instructions, processor 202 mayretrieve the instructions from the internal register or internal cacheand decode them. During or after execution of the instructions,processor 202 may write one or more results (which may be intermediateor final results) to the internal register or internal cache. Processor202 may then write one or more of those results to memory 204. Inparticular embodiments, processor 202 executes only instructions in oneor more internal registers or internal caches or in memory 204 (asopposed to storage 206 or elsewhere) and operates only on data in one ormore internal registers or internal caches or in memory 204 (as opposedto storage 206 or elsewhere). One or more memory buses (which may eachinclude an address bus and a data bus) may couple processor 202 tomemory 204. Bus 212 may include one or more memory buses, as describedbelow. In particular embodiments, one or more memory management units(MMUs) reside between processor 202 and memory 204 and facilitateaccesses to memory 204 requested by processor 202. In particularembodiments, memory 204 includes random access memory (RAM). This RAMmay be volatile memory, where appropriate Where appropriate, this RAMmay be dynamic RAM (DRAM) or static RAM (SRAM). Moreover, whereappropriate, this RAM may be single-ported or multi-ported RAM. Thisdisclosure contemplates any suitable RAM. Memory 204 may include one ormore memories 204, where appropriate. Although this disclosure describesand illustrates particular memory, this disclosure contemplates anysuitable memory.

In particular embodiments, storage 206 includes mass storage for data orinstructions. As an example and not by way of limitation, storage 206may include a hard disk drive (HDD), a floppy disk drive, flash memory,an optical disc, a magneto-optical disc, magnetic tape, or a UniversalSerial Bus (USB) drive or a combination of two or more of these. Storage206 may include removable or non-removable (or fixed) media, whereappropriate. Storage 206 may be internal or external to computer system200, where appropriate. In particular embodiments, storage 206 isnon-volatile, solid-state memory. In particular embodiments, storage 206includes read-only memory (ROM). Where appropriate, this ROM may bemask-programmed ROM, programmable ROM (PROM), erasable PROM (EPROM),electrically erasable PROM (EEPROM), electrically alterable ROM (EAROM),or flash memory or a combination of two or more of these. Thisdisclosure contemplates mass storage 206 taking any suitable physicalform. Storage 206 may include one or more storage control unitsfacilitating communication between processor 202 and storage 206, whereappropriate. Where appropriate, storage 206 may include one or morestorages 206. Although this disclosure describes and illustratesparticular storage, this disclosure contemplates any suitable storage.

In particular embodiments, I/O interface 208 includes hardware,software, or both, providing one or more interfaces for communicationbetween computer system 200 and one or more I/O devices. Computer system200 may include one or more of these I/O devices, where appropriate. Oneor more of these I/O devices may enable communication between a personand computer system 200. As an example and not by way of limitation, anI/O device may include a keyboard, keypad, microphone, monitor, mouse,printer, scanner, speaker, still camera, stylus, tablet, touch screen,trackball, video camera, another suitable I/O device or a combination oftwo or more of these. An I/O device may include one or more sensors.This disclosure contemplates any suitable I/O devices and any suitableI/O interfaces 208 for them. Where appropriate, I/O interface 208 mayinclude one or more device or software drivers enabling processor 202 todrive one or more of these I/O devices. I/O interface 208 may includeone or more I/O interfaces 208, where appropriate. Although thisdisclosure describes and illustrates a particular I/O interface, thisdisclosure contemplates any suitable I/O interface.

In particular embodiments, communication interface 210 includeshardware, software, or both providing one or more interfaces forcommunication (such as, for example, packet-based communication) betweencomputer system 200 and one or more other computer systems 200 or one ormore networks. As an example and not by way of limitation, communicationinterface 210 may include a network interface controller (NIC) ornetwork adapter for communicating with an Ethernet or other wire-basednetwork or a wireless NIC (WNIC) or wireless adapter for communicatingwith a wireless network, such as a WI-FI network. This disclosurecontemplates any suitable network and any suitable communicationinterface 210 for it. As an example and not by way of limitation,computer system 200 may communicate with an ad hoc network, a personalarea network (PAN), a local area network (LAN), a wide area network(WAN), a metropolitan area network (MAN), or one or more portions of theInternet or a combination of two or more of these. One or more portionsof one or more of these networks may be wired or wireless. As anexample, computer system 200 may communicate with a wireless PAN (WPAN)(such as, for example, a BLUETOOTH WPAN), a WI-FI network, a WI-MAXnetwork, a cellular telephone network (such as, for example, a GlobalSystem for Mobile Communications (GSM) network), or other suitablewireless network or a combination of two or more of these. Computersystem 200 may include any suitable communication interface 210 for anyof these networks, where appropriate. Communication interface 210 mayinclude one or more communication interfaces 210, where appropriate.Although this disclosure describes and illustrates a particularcommunication interface, this disclosure contemplates any suitablecommunication interface.

In particular embodiments, bus 212 includes hardware, software, or bothcoupling components of computer system 200 to each other. As an exampleand not by way of limitation, bus 212 may include an AcceleratedGraphics Port (AGP) or other graphics bus, an Enhanced Industry StandardArchitecture (EISA) bus, a front-side bus (FSB), a HYPERTRANSPORT (HT)interconnect, an Industry Standard Architecture (ISA) bus, an INFINIBANDinterconnect, a low-pin-count (LPC) bus, a memory bus, a Micro ChannelArchitecture (MCA) bus, a Peripheral Component Interconnect (PCI) bus, aPCI-Express (PCIe) bus, a serial advanced technology attachment (SATA)bus, a Video Electronics Standards Association local (VLB) bus, oranother suitable bus or a combination of two or more of these. Bus 212may include one or more buses 212, where appropriate. Although thisdisclosure describes and illustrates a particular bus, this disclosurecontemplates any suitable bus or interconnect.

Herein, a computer-readable non-transitory storage medium or media mayinclude one or more semiconductor-based or other integrated circuits(ICs) (such, as for example, field-programmable gate arrays (FPGAs) orapplication-specific ICs (ASICs)), hard disk drives (HDDs), hybrid harddrives (HHDs), optical discs, optical disc drives (ODDs),magneto-optical discs, magneto-optical drives, floppy diskettes, floppydisk drives (FDDs), magnetic tapes, solid-state drives (SSDs),RAM-drives, SECURE DIGITAL cards or drives, any other suitablecomputer-readable non-transitory storage media, or any suitablecombination of two or more of these, where appropriate. Acomputer-readable non-transitory storage medium may be volatile,non-volatile, or a combination of volatile and non-volatile, whereappropriate.

Herein, “or” is inclusive and not exclusive, unless expressly indicatedotherwise or indicated otherwise by context. Therefore, herein, “A or B”means “A, B, or both,” unless expressly indicated otherwise or indicatedotherwise by context. Moreover, “and” is both joint and several, unlessexpressly indicated otherwise or indicated otherwise by context.Therefore, herein, “A and B” means “A and B, jointly or severally,”unless expressly indicated otherwise or indicated otherwise by context.

The scope of this disclosure encompasses all changes, substitutions,variations, alterations, and modifications to the example embodimentsdescribed or illustrated herein that a person having ordinary skill inthe art would comprehend. The scope of this disclosure is not limited tothe example embodiments described or illustrated herein. Moreover,although this disclosure describes and illustrates respectiveembodiments herein as including particular components, elements,feature, functions, operations, or steps, any of these embodiments mayinclude any combination or permutation of any of the components,elements, features, functions, operations, or steps described orillustrated anywhere herein that a person having ordinary skill in theart would comprehend. Furthermore, reference in the appended claims toan apparatus or system or a component of an apparatus or system beingadapted to, arranged to, capable of, configured to, enabled to, operableto, or operative to perform a particular function encompasses thatapparatus, system, component, whether or not it or that particularfunction is activated, turned on, or unlocked, as long as thatapparatus, system, or component is so adapted, arranged, capable,configured, enabled, operable, or operative.

What is claimed is:
 1. An apparatus comprising: a touch sensorcomprising a mesh of conductive material, the mesh comprising aplurality of mesh cells that each have a plurality of vertices, each ofone or more of the vertices having a substantially randomized locationwithin an annulus centered at a seed location of the vertex; one or morecomputer-readable non-transitory storage media coupled to the touchsensor and embodying logic that is configured when executed to controlthe touch sensor.
 2. The apparatus of claim 1, wherein one or more ofthe seed locations are determined relative to a display underneath thetouch sensor, the display comprising a plurality of pixels that eachcomprise sub-pixels, each of the pixels having a first pixel pitch alonga first axis and a second pixel pitch along a second axis that isperpendicular to the first axis, each of the sub-pixels having a firstsub-pixel pitch along the first axis, a first sub-pixel dimension alongthe first axis, and a second sub-pixel dimension along the second axis.3. The apparatus of claim 2, wherein the vertices of each mesh cellcomprise: a terminal side of a first angle relative to the first axiscomprising a first seed location and a second seed location, the firstangle being at least approximately equal to an arctangent of the ratioof the second sub-pixel dimension to the first pixel pitch; and aterminal side of a second angle relative to the first axis comprisingthe second seed location and a third seed location, the second anglebeing at least approximately equal to an arctangent of the ratio of thesecond pixel pitch to twice the first sub-pixel pitch, wherein the firstand third seed locations being opposing seed locations of the mesh cell,the second seed location being adjacent to the first and third seedlocations.
 4. The apparatus of claim 3, wherein a distance separatingthe first seed location from the second seed location is at leastapproximately equal to three times the first pixel pitch and a distanceseparating the second seed location from the third seed location is atleast approximately equal to 13/6 of the first pixel pitch.
 5. Theapparatus of claim 3, wherein a distance separating the first seedlocation from the second seed location is at least approximately equalto six times the first pixel pitch; and a distance separating the secondseed location from the third seed location is at least approximatelyequal to 13/3 of the first pixel pitch.
 6. The apparatus of claim 3,wherein each of the sub-pixels having a first sub-pixel dead spacedimension along the first axis, a distance separating the first seedlocation from the second seed location is at least approximately equalto two times the first pixel pitch, a distance separating the secondseed location from the third seed location is at least approximatelyequal to a sum of: the first pixel pitch; the first sub-pixel pitch; andthe first sub-pixel dead space dimension; and a half of the firstsub-pixel dimension.
 7. The apparatus of claim 1, wherein the meshfurther comprises an initial mesh cell with four seed locations and seedlocations of a subsequent mesh cell are determined based at least inpart on the vertices of the initial mesh cell.
 8. The apparatus of claim1, wherein: the mesh further comprises a first subset of mesh cellsseparated from a second subset of mesh cells by at least a thickness ofa dielectric layer; and the seed location of each vertex of the secondsubset of mesh cells is within a centroid of each mesh cell of the firstsubset of mesh cells.
 9. The apparatus of claim 8, wherein one or moremesh segments coupling adjacent vertices of the second subset of meshcells overlap a midpoint of a mesh segment coupling adjacent vertices ofthe first subset of mesh cells.
 10. A touch sensor comprising: a mesh ofconductive material, the mesh comprising a plurality of mesh cells; eachof the mesh cells having a plurality of vertices; and each of one ormore of the vertices having a substantially randomized location withinan annulus centered at a seed location of the vertex;
 11. The touchsensor of claim 10, wherein one or more of the seed locations aredetermined relative to a display underneath the touch sensor, thedisplay comprising a plurality of pixels that each comprise sub-pixels,each of the pixels having a first pixel pitch along a first axis and asecond pixel pitch along a second axis that is perpendicular to thefirst axis, each of the sub-pixels having a first sub-pixel pitch alongthe first axis, a first sub-pixel dimension along the first axis, and asecond sub-pixel dimension along the second axis.
 12. The touch sensorof claim 11, wherein the vertices of each mesh cell comprise: a terminalside of a first angle relative to the first axis comprising a first seedlocation and a second seed location, the first angle being at leastapproximately equal to an arctangent of the ratio of the secondsub-pixel dimension to the first pixel pitch; and a terminal side of asecond angle relative to the first axis comprising the second seedlocation and a third seed location, the second angle being at leastapproximately equal to an arctangent of the ratio of the second pixelpitch to twice the first sub-pixel pitch, wherein the first and thirdseed locations being opposing seed locations of the mesh cell, thesecond seed location being adjacent to the first and third seedlocations.
 13. The touch sensor of claim 12, wherein a distanceseparating the first seed location from the second seed location is atleast approximately equal to three times the first pixel pitch and adistance separating the second seed location from the third seedlocation is at least approximately equal to 13/6 of the first pixelpitch.
 14. The touch sensor of claim 12, wherein a distance separatingthe first seed location from the second seed location is at leastapproximately equal to six times the first pixel pitch; and a distanceseparating the second seed location from the third seed location is atleast approximately equal to 13/3 of the first pixel pitch.
 15. Thetouch sensor of claim 12, wherein each of the sub-pixels having a firstsub-pixel dead space dimension along the first axis, a distanceseparating the first seed location from the second seed location is atleast approximately equal to two times the first pixel pitch, a distanceseparating the second seed location from the third seed location is atleast approximately equal to a sum of: the first pixel pitch; the firstsub-pixel pitch; and the first sub-pixel dead space dimension; and ahalf of the first sub-pixel dimension.
 16. The touch sensor of claim 10,wherein the mesh further comprises an initial mesh cell with four seedlocations and seed locations of a subsequent mesh cell are determinedbased at least in part on the vertices of the initial mesh cell.
 17. Thetouch sensor of claim 10, wherein: the mesh further comprises a firstsubset of mesh cells separated from a second subset of mesh cells by atleast a thickness of a dielectric layer; and the seed location of eachvertex of the second subset of mesh cells is within a centroid of eachmesh cell of the first subset of mesh cells.
 18. The touch sensor ofclaim 17, wherein one or more mesh segments coupling adjacent verticesof the second subset of mesh cells overlap a midpoint of a mesh segmentcoupling adjacent vertices of the first subset of mesh cells.
 19. Amethod comprising: by a computing device, determining a plurality ofseed locations; and by the computing device, generating a pattern for amesh of conductive material of a touch sensor at least in part bydetermining a plurality of vertices of a plurality of mesh cells of themesh of conductive material, each of one or more of the vertices havinga substantially randomized location within an annulus centered at one ofthe seed locations.
 20. The method of claim 19, wherein one or more ofthe seed locations are determined relative to a display underneath thetouch sensor, the display comprising a plurality of pixels that eachcomprise sub-pixels, each of the pixels having a first pixel pitch alonga first axis and a second pixel pitch along a second axis that isperpendicular to the first axis, each of the sub-pixels having a firstsub-pixel pitch along the first axis, a first sub-pixel dimension alongthe first axis, and a second sub-pixel dimension along the second axis,and wherein the vertices of each mesh cell comprise a terminal side of afirst angle relative to the first axis comprising a first seed locationand a second seed location, the first angle being at least approximatelyequal to an arctangent of the ratio of the second sub-pixel dimension tothe first pixel pitch, a terminal side of a second angle relative to thefirst axis comprising the second seed location and a third seedlocation, the second angle being at least approximately equal to anarctangent of the ratio of the second pixel pitch to twice the firstsub-pixel pitch, wherein the first and third seed locations beingopposing seed locations of the mesh cell, the second seed location beingadjacent to the first and third seed locations.